Nrbox System Description for CoCo 2016 Takahito Aoto (Niigata - PowerPoint PPT Presentation

Nrbox System Description for CoCo 2016 Takahito Aoto (Niigata University) Kentaro Kikuchi (Tohoku University)

nrbox : n ominal r ewriting tool box Confluence Tool for “ Nominal Rewriting Systems ” ◮ Written in SML/NJ, about 4500 loc ◮ http://www.nue.ie.niigata-u.ac.jp/tools/nrbox/ ◮ Requires an external termination prover for first-order TRSs. ◮ Entrant for Demo(NRS) category. 1 / 4

Nominal Rewriting (Fernández & Gabbay, 2007) ◮ Extension of first-order term rewriting ◮ Binding mechanism ◮ Nominal approach (Gabbay & Pitts, 2002) ◮ α -equivalence is dealt with at object-level ◮ In contrast to traditional higher-order rewriting, which uses λ -calculus as meta-calculus. 2 / 4

Nominal Rewriting (Fernández & Gabbay, 2007) ◮ Extension of first-order term rewriting ◮ Binding mechanism ◮ Nominal approach (Gabbay & Pitts, 2002) ◮ α -equivalence is dealt with at object-level ◮ In contrast to traditional higher-order rewriting, which uses λ -calculus as meta-calculus. Example. ( ∀ a.P ) ∧ Q ≡ ∀ a. ( P ∧ Q ) ( a / ∈ FV ( Q )) P ∧ ( ∀ a.Q ) ≡ ∀ a. ( P ∧ Q ) ( a / ∈ FV ( P )) Nominal rewriting system (NRS for short) � a # Q ⊢ and ( forall ([ a ] P ) , Q ) → forall ([ a ] and ( P, Q )) a # P ⊢ and ( P, forall ([ a ] Q )) → forall ([ a ] and ( P, Q )) 2 / 4

Rewriting by NRSs Definition [rewrite relation] def ⇒ ∆ ⊢ ∇ π σ, ∆ ⊢ s | p ≈ α l π σ, t = s [ r π σ ] p . ∆ ⊢ s → � R,π,p,σ � t ⇐ def ∆ ⊢ s → R t ⇐ ⇒ ∆ ⊢ s → � R,π,p,σ � t for some R ∈ R , π, p, σ . and ( forall ([ a ] p ( a )) , forall ([ a ] q ( a ))) → R 1 forall ([ a ] and ( p ( a ) , forall ([ a ] q ( a )))) ≈ α forall ([ a ] and ( p ( a ) , forall ([ b ] q ( b )))) → R ( a b ) forall ([ a ] forall ([ b ] and ( p ( a ) , q ( b )))) 2 3 / 4

Rewriting by NRSs Definition [rewrite relation] def ⇒ ∆ ⊢ ∇ π σ, ∆ ⊢ s | p ≈ α l π σ, t = s [ r π σ ] p . ∆ ⊢ s → � R,π,p,σ � t ⇐ def ∆ ⊢ s → R t ⇐ ⇒ ∆ ⊢ s → � R,π,p,σ � t for some R ∈ R , π, p, σ . and ( forall ([ a ] p ( a )) , forall ([ a ] q ( a ))) → R 1 forall ([ a ] and ( p ( a ) , forall ([ a ] q ( a )))) ≈ α forall ([ a ] and ( p ( a ) , forall ([ b ] q ( b )))) → R ( a b ) forall ([ a ] forall ([ b ] and ( p ( a ) , q ( b )))) 2 Definition [Church-Rosser modulo ≈ α (CRM for short)] ( ← ∪ → ∪ ≈ α ) ∗ t 1 t 2 * * s 1 s 2 ≈ α 3 / 4

Implemented Confluence Criteria Proposition [SKAT, RTA 2015] Abstract skeleton preserving orthogonal NRSs are CRM. Proposition [SKAT, SCSS 2016] Linear uniform NRSs are CRM if Γ ⊢ u → = ◦ ≈ α ◦ ← ∗ v and Γ ⊢ u → ∗ ◦ ≈ α ◦ ← = v for any BCP Γ ⊢ � u, v � . Proposition [SKAT, SCSS 2016] Terminating uniform NRSs are CRM if and only if Γ ⊢ u ( → ∗ ◦ ≈ α ◦ ← ∗ ) v for any BCP Γ ⊢ � u, v � . Proposition [KAT, 2016] Left-linear uniform NRSs are CRM if Γ ⊢ u − → ◦ ≈ α v � ◦ ≈ α ◦ ← ∗ v ) for any inner (resp. outer) BCP Γ ⊢ � u, v � . ( u − → � 4 / 4